XII. Elementary Function

Transcription

1 XII. Elementary Function Yuxi Fu BASICS, Shanghai Jiao Tong University

2 What is the class of arithmetic functions we use in mathematics? Computability Theory, by Y. Fu XII. Elementary Function 1 / 17

3 Definition The class E of elementary function is constructed from 1. zero, successor, projection, and 2. subtraction x y (for defining conditionals); and is 3. closed under composition and 4. bounded sum/product (bounded recursion). Remark. 1. Addition and multiplication can be defined using bounded sum; (hyper) exponential can be defined using bounded product. 2. Lower elementary functions are constructed by dropping (4). 3. A predicate is elementary if its characterization function is elementary. Computability Theory, by Y. Fu XII. Elementary Function 2 / 17

4 Bounded Minimalisation is Elementary Fact. E is closed under bounded minimalisation. Proof. Suppose f (x, z) is elementary. Then µz < y.f (x, z) = 0 is sg(f (x, u)). v<y u v It is easy to see that sg is elementary. Computability Theory, by Y. Fu XII. Elementary Function 3 / 17

5 Logical Operation Fact. The set of elementary predicates is closed under negation, conjunction, disjunction and bounded quantifiers. Computability Theory, by Y. Fu XII. Elementary Function 4 / 17

6 Basic Arithmetic Functions are Elementary 1. The exponential x y is defined by z<y U2 1 (x, z). 2. The function p x is defined by p x = µy < 2 2x.(x = 0 or y is the xth prime) = µy < 2 2x. x = Pr(z) z y = µy < 2 2x. x Pr(z) = 0. z y Computability Theory, by Y. Fu XII. Elementary Function 5 / 17

7 Bounded Recursion Fact. Let f (x) and g(x, y, z) be elementary and h(x, y) be defined from f, g via primitive recursion. If h(x, y) b(x, y) for some elementary function b(x, y), then h(x, y) is elementary. Proof. Observe that 2 h(x,0) 3 h(x,1)... p h(x,y) y+1 p b(x,z) z+1. z y So we can define h(x, y) by µe z y p b(x,z) z+1. ((e) 1 = f (x) z < y.((e) z+2 = g(x, z, (e) z+1 ))). We are done. Computability Theory, by Y. Fu XII. Elementary Function 6 / 17

8 Gödel Encoding is Elementary Fact. Gödel encoding functions are elementary. Computability Theory, by Y. Fu XII. Elementary Function 7 / 17

9 Kleene s Predicate is Elementary Fact. The Kleene s functions σ n, c n and j n are elementary. Computability Theory, by Y. Fu XII. Elementary Function 8 / 17

10 Elementary Time Functions A computable function φ e is in elementary time if t e (x) b(x) almost everywhere for some elementary function b(x). Fact. The elementary time functions are elementary. Proof. φ e (x) is almost everywhere computable by the elementary function (c n (e, x, µt b( x).j n (e, x, t) = 0)) 1, which implies that φ e (x) is elementary. Computability Theory, by Y. Fu XII. Elementary Function 9 / 17

11 It has been suggested that E contains all practical computable functions. Computability Theory, by Y. Fu XII. Elementary Function 10 / 17

12 A computable function f (x) is practically computable if it can be computed in exp k (x) = x }{{} k steps for some k. We let 2 exp k(x) stand for exp k+1 (x). Computability Theory, by Y. Fu XII. Elementary Function 11 / 17

13 Upper Bound of Elementary Functions Theorem. For each elementary function f ( x) there is some k such that f ( x) exp k (max{ x}). Proof. The basic elementary functions satisfy the upper bound. The elementary operations preserves the upper bound. Computability Theory, by Y. Fu XII. Elementary Function 12 / 17

14 Corollary. exp x (x) is primitive recursive but not elementary. Proof. The function exp x (x) is defined by g(x, x), where We are done. g(x, 0) = x, g(x, y + 1) = 2 g(x,y). Computability Theory, by Y. Fu XII. Elementary Function 13 / 17

15 Elementary Functions are Elementary Time Lemma. Suppose f ( x) and g( x, y, z) are in elementary time and h( x, y) is defined from f, g via recursion. If h( x, y) is elementary, then h( x, y) is in elementary time. Proof. The standard program that calculates h does it in elementary time. Computability Theory, by Y. Fu XII. Elementary Function 14 / 17

16 Elementary Functions are Elementary Time Theorem. If f ( x) is elementary, then there is a program P for f such that tp n ( x) is elementary. Proof. Use the above lemma. Computability Theory, by Y. Fu XII. Elementary Function 15 / 17

17 Complexity Theoretical Characterization Theorem. A total function f ( x) is elementary iff it is computable in time exp k (max{ x}) for some k. Computability Theory, by Y. Fu XII. Elementary Function 16 / 17

18 ELEMENTARY = TIME(2 n ) TIME(2 2n ) TIME(2 22n ).... Computability Theory, by Y. Fu XII. Elementary Function 17 / 17